Rewriting expressions with exponents challenge 1 | Algebra 1 (TX TEKS) | Khan Academy

- [Presenter] So we have
this pretty complicated,

some would say hairy,
expression right over here.

And what I want you to
do is pause this video

and see if you can simplify this

based on what you know
about exponent rules.

All right, now let's do this together.

There's many ways you could approach this,

but what my brain wants to do is first

try to simplify this part right over here.

I have a bunch of stuff in
here to an exponent power.

And one way to think about
that, if I have, let's say,

A times B

to the, let's call it C, power.

This is the same thing
as A to the C times B

to the C power.

So we could do that with
this part right over here.

And actually, let me just simplify this

so I don't have to keep rewriting things.

So this can be rewritten as

5M, or let me be careful.

This is gonna be five squared

times M to the negative one third squared

times N squared,

which is the same thing as 25.

Now if I raise something to an exponent

and then raise that to an exponent,

so there's another exponent property here.

If I have A to the B, and
then I raise that to the C,

then I multiplied the exponents.

This is equal to A to the B times C power.

So here we would multiply these exponents.

So it's 25M.

Two times negative one third
is negative two thirds.

And then of course we have
this N squared right over here.

So actually lemme just rewrite everything

so we don't lose too much track.

So we have 75.

I wrote M. 75.

M to the one third. N
to the negative seven.

And then I simplified the bottom part.

I'll do that same color. As 25.

M to the negative two thirds

N squared.

Now, some of y'all might
immediately be able

to skip some steps here, but I'll try

to make it very, very explicit.

What I'm going to read, what I'm gonna do

is rewrite this expression
as the product of fractions

or as a product of rational expression.

So I could rewrite this.

This is being equal to

75 divided by 25,

which I think you know what that is.

But I'll just write it like that.

Times, and then we'll worry
about these right over here.

Times M to the one third

over M to the negative two thirds,

and then times, I'll put this in blue.

Times N to the negative seven

over N squared.

Now 75 over 25, we know what that is,

that's going to be equal to three.

But how do we simplify
this right over here?

Well, here we can remind ourselves

another exponent property.

If I have, let's call it A to the B

over C to the D, actually it
has to have the same base.

Over A to the C.

This is going to be the same thing

as A to the B minus C power.

So I can rewrite all of this business.

I have my three here.

Three times

M to the one third.

And then I'm gonna subtract this exponent.

We have to be very careful.
We're subtracting a negative.

So we're subtracting negative two thirds.

That's all that exponent for M.

And then we're going to have times N

to the negative seven power

minus two.

And so now we are in the home stretch.

This is going to be equal to

three times M to the,

what's one third minus
negative two thirds?

Well, that's the same thing
as one third plus two thirds,

which is just three
thirds, which is just one.

So this is just M to the first power,

which is the same thing as just M.

And then that is going to be times

negative seven minus two,
that is negative nine.

So times N to the negative
ninth power. And we are done.

And that is strangely
satisfying to take something

that hairy and make it,
I guess, less hairy?

Now, some folks might not like

having a negative nine exponent here.

They might want only positive exponents.

So you could actually rewrite
this and we could debate

whether it's actually
simpler or less simple.

But we also know the exponent properties

that if I have A to the negative N,

that is the same thing
as one over A to the N.

So based on that, I could
also rewrite this as three.

We do the same color as that three.

As three times M.

And then instead of saying
times N to the negative nine,

we could say that is over,

that is over, N to the ninth.

So that's another way to
rewrite that expression.